Lemma 21.20.8. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be an object of $\mathcal{C}$. Denote $j : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U), \mathcal{O}_ U) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ the corresponding localization morphism. The restriction functor $D(\mathcal{O}) \to D(\mathcal{O}_ U)$ is a right adjoint to extension by zero $j_! : D(\mathcal{O}_ U) \to D(\mathcal{O})$.
Proof. We have to show that
\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O})}(j_!E, F) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(E, F|_ U) \]
Choose a complex $\mathcal{E}^\bullet $ of $\mathcal{O}_ U$-modules representing $E$ and choose a K-injective complex $\mathcal{I}^\bullet $ representing $F$. By Lemma 21.20.1 the complex $\mathcal{I}^\bullet |_ U$ is K-injective as well. Hence we see that the formula above becomes
\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O})}(j_!\mathcal{E}^\bullet , \mathcal{I}^\bullet ) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(\mathcal{E}^\bullet , \mathcal{I}^\bullet |_ U) \]
which holds as $|_ U$ and $j_!$ are adjoint functors (Modules on Sites, Lemma 18.19.2) and Derived Categories, Lemma 13.31.2. $\square$
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